The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
Fermat Contest
(Grade 11)
Thursday, February 20, 2014
(in North America and South America)
Friday, February 21, 2014
(outside of North America and South America)
Time: 60 minutes
Calculators are permitted
Instructions
©2013 University of Waterloo
1. Do not open the Contest booklet until you are told to do so.
2. You may use rulers, compasses and paper for rough work.
3. Be sure that you understand the coding system for your response form. If you are not sure,
ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in
circles completely.
4. On your response form, print your school name and city/town in the box in the upper right
corner.
5. Be certain that you code your name, age, sex, grade, and the Contest you are
writing in the response form. Only those who do so can be counted as eligible
students.
6. This is a multiple-choice test. Each question is followed by five possible answers marked
A, B, C, D, and E. Only one of these is correct. After making your choice, fill in the
appropriate circle on the response form.
7. Scoring: Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
There is no penalty for an incorrect answer.
Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
8. Diagrams are not drawn to scale. They are intended as aids only.
9. When your supervisor tells you to begin, you will have sixty minutes of working time.
Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be
published on our website, http://www.cemc.uwaterloo.ca. In addition, the name, grade, school
and location, and score of some top-scoring students may be shared with other mathematical
organizations for other recognition opportunities.
Scoring:
There is no penalty for an incorrect answer.
Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
Part A: Each correct answer is worth 5.
1.
(A) 2
2.
15 − 32
?
3
(B) 4
What is the value of
(B) 101 and 102
(E) 104 and 105
(C) 102 and 103
(B) 4
(C) 0
(E) −4
(D) 12
Two positive integers x and y have xy = 24 and x − y = 5. The value of x + y is
(A) 10
5.
(E) 3
If x = 2, then (x + 2 − x)(2 − x − 2) equals
(A) −12
4.
(D) 48
The integer 2014 is between
(A) 100 and 101
(D) 103 and 104
3.
(C) 12
(B) 11
(C) 12
(D) 13
(E) 14
In the diagram, square W XY Z has area 9 and W is at
the centre of a circle. If X and Z are on the circle, the
area of the circle is
(A) 3π
(B) 6π
(D) 18π
(E) 81π
Y
X
(C) 9π
Z
W
6.
If 50% of N is 16, then 75% of N is
(A) 12
7.
(B) 6
(C) 20
(D) 24
(E) 40
In the diagram, point T is on side P R of 4P QR and
QRS is a straight line segment. The value of x is
(A) 55
(B) 70
(D) 60
(E) 50
P
(C) 75
30˚
T
S
(180 – x)˚
R
x˚
x˚
Q
8.
In a group of five friends:
• Amy is taller than Carla.
• Dan is shorter than Eric but taller than Bob.
• Eric is shorter than Carla.
Who is the shortest?
(A) Amy
9.
(B) Bob
(C) Carla
(D) Dan
In the diagram, P QRS is a square with side length 8.
Points T and U are on P S and QR respectively with
QU = T S = 1. The length of T U is closest to
(A) 8.5
(B) 9.9
(D) 10.6
(E) 11.3
(E) Eric
P
Q
U
(C) 10
T
S
R
10. A line segment of length 5 lies along the number line initially between 1 and 6.
–3 –2 –1
0
1
2
3
4
5
6
7
8
The line segment is rotated 180◦ about the point at 2. The resulting line segment
lies between −2 and 3. This segment is then rotated 180◦ about the point at 1. The
resulting line segment lies between
(A) −2 and 3
(B) −1 and 4
(C) 0 and 5
(D) −3 and 2
(E) −4 and 1
Part B: Each correct answer is worth 6.
2
9a + 8b
11. If a = b and b 6= 0, then
is equal to
3
6a
7
3
(B) 9
(C)
(A)
2
2
(D)
11
2
(E)
17
6
(D)
4
5
(E) 3
12. If 10x · 105 = 1004 , what is the value of x?
(A) 1
(B) 35
(C) 11
13. How many positive integers n between 10 and 1000 have the property that the sum
of the digits of n is 3?
(A) 10
(B) 8
(C) 6
(D) 9
(E) 7
14. Last summer, Pat worked at a summer camp.
For each day that he worked, he earned $100 and he was not charged for food.
For each day that he did not work, he was not paid and he was charged $20 for food.
After 70 days, the money that he earned minus his food costs equalled $5440.
On how many of these 70 days did Pat work?
(A) 60
(B) 68
(C) 50
(D) 57
(E) 34
15. On each spin of the spinner shown, the arrow is equally
likely to stop on any one of the four numbers. Deanna
spins the arrow on the spinner twice. She multiplies
together the two numbers on which the arrow stops.
Which product is most likely to occur?
(A) 2
(B) 4
(D) 8
(E) 12
4
1
3
2
(C) 6
16. At the start of a 5 hour trip, the odometer in Jill’s car indicates that her car had
already been driven 13 831 km. The integer 13 831 is a palindrome, because it is the
same when read forwards or backwards. At the end of the 5 hour trip, the odometer
reading was another palindrome. If Jill never drove faster than 80 km/h, her greatest
possible average speed was closest to
(A) 62 km/h
(B) 20 km/h
(C) 42 km/h
(D) 16 km/h
(E) 77 km/h
17. Sergio recently opened a store. One day, he determined that the average number of
items sold per employee to date was 75. The next day, one employee sold 6 items,
one employee sold 5 items, and one employee sold 4 items. The remaining employees
each sold 3 items. This made the new average number of items sold per employee to
date equal to 78.3. How many employees are there at the store?
(A) 50
(B) 5
(C) 20
(D) 40
(E) 30
18. A square is cut along a diagonal and reassembled to form
the parallelogram P QRS as shown in the diagram. If
P R = 90 mm, what is the area of the original square,
in mm2 ?
(A) 324
(B) 1620
(D) 2025
(E) 2700
S
R
(C) 1800
P
Q
19. Max and Minnie each add up sets of three-digit positive integers. Each of them adds
three different three-digit integers whose nine digits are all different. Max creates
the largest possible sum. Minnie creates the smallest possible sum. The difference
between Max’s sum and Minnie’s sum is
(A) 594
(B) 1782
(C) 1845
(D) 1521
(E) 2592
20. In the diagram, 4P QR has P Q = QR = RP = 30.
Points S and T are on P Q and P R, respectively, so that
ST is parallel to QR. Points V and U are on QR so
that T U is parallel to P Q and SV is parallel to P R. If
V S + ST + T U = 35, the length of V U is
(A) 21
(B) 15
(D) 20
(E) 25
P
(C) 18
S
Q
T
V
U
R
Part C: Each correct answer is worth 8.
21. A bin contains 10 kg of peanuts. 2 kg of peanuts are removed and 2 kg of raisins are
added and thoroughly mixed in. Then 2 kg of this mixture are removed and 2 kg of
raisins are added and thoroughly mixed in again. What is the ratio of the mass of
peanuts to the mass of raisins in the final mixture?
(A) 3 : 2
(B) 4 : 1
(C) 5 : 1
(D) 7 : 3
(E) 16 : 9
22. Jillian drives along a straight road that goes directly from her house (J) to her
Grandfather’s house (G). Some of this road is on flat ground and some is downhill
or uphill. Her car travels downhill at 99 km/h, on flat ground at 77 km/h, and uphill
at 63 km/h. It takes Jillian 3 hours and 40 minutes to drive from J to G. It takes
her 4 hours and 20 minutes to drive from G to J. The distance between J and G, in
km, is
(A) 318 32
(B) 324
(C) 308
(D) 292 53
(E) 292 19
23. 4P QR has P Q = 150 and P R = QR = 125, as shown.
Three line segments are drawn parallel to QR, dividing
4P QR into four sections of equal area. The height, h,
of the bottom section is closest to
(A) 16.7
(B) 16.9
(D) 16.3
(E) 16.1
P
(C) 16.5
h
Q
R
24. Mohammed has eight boxes numbered 1 to 8 and eight balls numbered 1 to 8. In how
many ways can he put the balls in the boxes so that there is one ball in each box,
ball 1 is not in box 1, ball 2 is not in box 2, and ball 3 is not in box 3?
(A) 27 240
(B) 29 160
(C) 27 360
(D) 27 600
(E) 25 200
25. Points P (r, s) and Q(t, u) are on the parabola with equation y = x2 − 51 mx + 15 n so
that P Q = 13 and the slope of P Q is 12
5 . For how many pairs (m, n) of positive
integers with n ≤ 1000 is r + s + t + u = 27 ?
(A) 28
(B) 26
(C) 27
(D) 29
(E) 25
2014
Fermat
Contest
(English)
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
For students...
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